Chaplin - Nuclear Interactions

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NUCLEAR ENERGY MATERIALS AND REACTORS Nuclear Interactions - R.A. Chaplin

NUCLEAR INTERACTIONS

R.A. Chaplin

University of New Brunswick, Canada

Keywords: Nuclear Interactions, Nuclear Cross Sections, Neutron Energies, Fission

and Fusion

Contents

1. Neutron Interactions

2. Nuclear Cross Sections

3. Neutron Scattering and Capture

4. Neutron Moderation

5. Fission and FusionGlossary

Bibliography

Biographical Sketch

To cite this chapter

Summary

When a heavy element, such as uranium, fissions into two mid range elements, binding

energy is released. Furthermore since the neutron to proton ratio is about 1.5 for the

heaviest elements but in the range of 1.2 to 1.3 for mid range elements there is a surplus of

neutrons after a fissioning process. Some heavy elements fission spontaneously at a veryslow rate due to inherent instability. However fissioning can be induced by adding energy

to the nucleus of some elements. This can be done by allowing the nucleus to capture a

free neutron which then adds sufficient binding energy, as it combines with the nucleus, to

cause the nucleus to become highly unstable and to split into two parts with additional free

neutrons. These components fly apart with high kinetic energy which is subsequently

degraded to produce heat.

Free neutrons interact with the nuclei of other materials in various ways, the most common

being absorption and scattering. Scattering results in the transfer of some energy and the

neutron continues to move through the medium but at a lower energy and hence lower

velocity. Neutrons being uncharged do not interact with the electron cloud surrounding the

nucleus and, since the nucleus occupies such a tiny space within the atom, the probability

of interaction is quite low. This probability is not necessarily related to the size of the

nucleus but is measured as a cross section in units of area. The cross sections of different

nuclei vary widely and may be greater or smaller than the projected area of the nucleus

itself.

To maintain an ongoing chain reaction of nuclear fissions to release energy at least one free

neutron from a previous fission must go on to induce fission in another fissile element such

as uranium. The probability of this occurring can be enhanced by reducing the velocity of

the neutron so that, when encountering a fissile nucleus, it spends more time in the

Encyclopedia of Life Support Systems (EOLSS)


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immediate vicinity of the nucleus. Thus surplus neutrons produced by fission are made to

pass through a suitable medium, known as a moderator, where their velocity is reduced by

multiple scattering collisions with moderator nuclei. They then re-enter the fissile fuel to

produce at least one further fission. Some neutrons are absorbed by various nuclei. This

process is carefully balanced to ensure the steady and continuous release of energy. Sinceonly about 200 MeV or 32 pJ is released by each fission, many parallel processes as

described above must occur simultaneously.

1. Neutron Interactions

1.1. Neutron Production

Neutrons can be created by the integration of an electron and a proton. Furthermore a free

neutron will in time disintegrate into a proton and an electron. Neutrons interact with the

nuclei of atoms in various ways and may also be produced by the nuclei of certain atoms.

The most common source of neutrons is the fissioning process where a heavy nucleus splits

into two lighter nuclei. This fissioning of nuclei and the subsequent interaction of the

resultant neutrons with other nuclei are the fundamental processes governing the

production of power from nuclear energy. Knowledge of these processes is important in

the study of nuclear engineering.

A heavy nucleus such as Uranium-235 will occasionally fission spontaneously into two

lighter nuclei. A heavy nucleus such as this has about one and a half as many neutrons as

protons in the nucleus. A mid-range nucleus however has only about one and a third as

many neutrons as protons in its nucleus. Thus, when a heavy nucleus fissions into two

lighter nuclei, not as many neutrons are required to maintain a stable configuration in thenucleus and some neutrons are rejected immediately the fission occurs. Generally two to

three neutrons are emitted during the fission process.

In a nuclear reactor, fissile nuclei such as Uranium-235 and Plutonium-239 are induced to

fission by having their nuclei excited beyond the level of stability. This is done by

subjecting them to the influence of free neutrons. Free neutrons interact with various

nuclei in different ways causing a range of different reactions of which fission is just one.

Most interactions involve scattering (non-absorption) or capture (absorption) of the

neutrons and a transfer of energy. These reactions are important in maintaining and

controlling the fission reactions in nuclear reactors.

1.2. Elastic Scattering (Elastic Collision)

Elastic scattering occurs when a neutron strikes a nucleus and rebounds elastically. In such

a collision kinetic energy is transmitted elastically in accordance with the basic laws of

motion. If the nucleus is of the same mass as the neutron then a large amount of kinetic

energy is transferred to the nucleus. If the nucleus is of a much greater mass than the

neutron then most of the kinetic energy is retained by the neutron as it rebounds. The

amount of kinetic energy transferred also depends upon the angle of impact and hence the

direction of motion of the neutron and nucleus after the impact.



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1.3. Inelastic Scattering (Inelastic Collision)

Inelastic scattering occurs when a neutron strikes and enters a nucleus. The nucleus is

excited into an unstable condition and a neutron is immediately emitted but with a lower

energy than that of the entering neutron. The surplus energy is transferred to the nucleus askinetic energy and excitation energy. The excited nucleus subsequently returns to the

ground state by the emission of a -ray. Such collisions are inelastic since all the initial

kinetic energy does not reappear as kinetic energy. Some is absorbed by the nucleus and

subsequently emitted in a different form ( -ray). The emitted neutron may or may not be

the one that initially struck the nucleus. In simplistic terms the neutron can be considered

simply to be bouncing off an energy absorbing nucleus.

1.4. Radiative Capture

Radiative capture can be considered to be similar to the initial process leading to inelasticscattering. A neutron strikes and enters a nucleus. The nucleus is excited but the level of

excitation is insufficient to eject a neutron. Instead all the energy is transferred to the

nucleus as kinetic energy and excitation energy. The excited nucleus subsequently returns

to the ground state by the emission of a -ray. The incoming neutron remains in the

nucleus and the nuclide increases its number of neutrons by one. This is a very common

type of reaction. It leads to the creation of heavier isotopes of the original element. Many

of these may be radioactive and decay over time in different ways.

1.5. Nuclear Transmutation (Charged Particle Reaction)

Nuclear transmutation is similar to radiative capture and inelastic scattering. A neutronstrikes and enters a nucleus. The nucleus is excited into an unstable condition but a particle

other than a neutron is emitted. The emitted particles are either protons or -particles.

This leaves the nucleus still in an excited state and it subsequently returns to the ground

state by the emission of a -ray. In this process the total number of protons in the nucleus

is reduced by one for proton emission and by two for -particle emission. The original

element is thus changed or transmuted into a different element.

1.6. Neutron Producing Reaction

Neutron producing reactions occur when one or two additional neutrons are produced froma single neutron. As before a neutron strikes and enters a nucleus. The nucleus is excited

into an unstable condition as with inelastic scattering but two or three neutrons instead of

only one neutron are emitted. The still excited nucleus subsequently returns to its ground

state by the emission of a -ray. This is an uncommon reaction occurring in only a few

isotopes.

1.7. Fission

Although spontaneous fission occasionally occurs, fission is generally induced by neutrons.

A neutron strikes and enters a heavy nucleus. The nucleus is excited into an unstable

condition as with most of the foregoing interactions. In this unstable condition the nucleus



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splits into two new mid-range nuclei usually of unequal mass. Since these new nuclei do

not need as many neutrons for stability some neutrons are emitted immediately. The

surplus binding energy drives the new nuclei (fission fragments) and neutrons away from

one another with high velocity. The new nuclei subsequently lose their kinetic energy by

ionizing reactions with the surrounding nuclei through which they pass and return to theirground states by emission of -rays. They are invariably still unstable with too many

neutrons and subsequently decay usually by -particle and

-ray emission. The high

energy neutrons lose energy by scattering collisions with nuclei of the surrounding medium

and are subsequently generally captured by other nuclei to produce one of the reactions

described in this section.

1.8. Neutron Flux

Neutrons created by fission pass freely through solid material since atoms consist mainly of

empty space. They have no charge and so are not affected by the charged electron cloudsurrounding the nucleus. Furthermore the nucleus is so small compared with the size of the

atom that the chance of the neutron colliding with it is extremely small. In a uniform

material the neutrons travel randomly in all directions and some measure of their number or

influence is required. A convenient parameter is neutron flux.

Neutron flux is defined as the number of neutrons per unit volume multiplied by their

velocity .

n

v

nv = (1)

Neutron flux so defined has units of number per unit area per unit time. This can be

considered as the number of neutrons passing through a particular cross sectional area in

any direction per second.

If the neutrons travel in a parallel beam the area through which the neutrons pass may be

considered to be at right angles to the beam and the given area will then be equal to the

cross sectional area of the beam. This is the case in irradiation experiments where a beam

of neutrons is directed out of a nuclear reactor through special ports which trap neutrons

moving in other directions. Such a beam is known as a collimated beam.

Within the reactor the neutrons travel in all directions and the neutrons will pass through agiven area in all directions and from both sides. This area is more difficult to define hence

the definition of neutron flux as number multiplied by velocity.

1.9. Neutron Energy

During the fission process, in which a heavy nucleus splits into two fission fragments and

some residual neutrons, some 200 MeV of binding energy is released. This appears as

kinetic energy as the fragments and neutrons separate at high velocity. Most energy is

carried by the fission fragments and is deposited as heat in the surrounding material as the

fragments come to rest. The two or three residual or prompt neutrons carry away about 5

MeV as kinetic energy so on average a neutron produced by fission has an energy of about



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2 MeV or 0.32 x 10-12

J. Considering that the mass of a neutron is 1.67495 x 10-27

kg its

velocity can be calculated from the basic equation for kinetic energy where m is mass

and V velocity:

KEE

2

KE E mV= (2)

This gives an average velocity of about 20 x 106

m/s. This is the average based on an

average energy of 2 MeV. The actual range of energies however can range from near zero

to about 8 MeV as shown in Figure 12 giving velocities anywhere up to about 55 x 106

m/s.

These high energy neutrons interact with the nuclei of the medium through which they

pass. In the process some are captured but most are scattered by elastic or inelastic

collisions with the nuclei. Such scattering collisions result in a transfer of energy from the

neutrons to the nuclei until the neutrons reach an equilibrium condition with the medium.

In this condition the nuclei, being in a state of vibratory motion by virtue of theirtemperature, give as much energy to the neutrons as they receive from the neutrons. The

neutrons are thus in thermal equilibrium with the medium and are said to be thermalized.

Even though the medium may be at a uniform temperature, subsequent scattering collisions

occurring in random directions relative to the motion of the nuclei, result in thermal

neutrons having a range of energies above and below the thermalization energy as shown in

Figure 1. This figure also shows the corresponding velocity distribution of the neutrons.

Figure 1. Energy and velocity distribution of thermalized neutrons



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This is a Maxwellian distribution with the energy E given in terms of the Boltzmann

constant k and temperature T as well as in electron-volts while the velocity V is given in

meters per second. The Boltzmann constant is as follows:

2413.8 10 J/Kk = 686.2 10 eV/Kk =

The average energy and the most probable energyaveE mpE of the neutrons are given by:

ave

mp

(3/2)

E kT

E kT

=

=

In neutron studies however the most probable velocity is considered. This is given by:mp

V

1/ 2

mp [2 / ]V kT m=

Hence the corresponding neutron energy E is given by:

E kT= (3)

All thermal neutrons in a system are considered to have this velocity which is then given

by:

2 mV kT =

At an ambient temperature of 20C or 293K this velocity is 2200 m/s and the

corresponding energy is 0.025 eV. These are the values traditionally used in neutron

scattering calculations involving thermal neutrons.

2. Nuclear Cross Sections

2.1. Microscopic Cross Sections

A solid material may be considered as being made up of tiny nuclei suspended in empty

space (the electron cloud of negligible mass). Each nucleus has an imaginary projected

area which may interfere with the passage of a neutron. A neutron entering the solid will

see these projected areas scattered everywhere but they are so small and so far apart (as

seen by the neutron) that the chances of hitting one is practically nil. Eventually a neutron

may hit a nucleus and will then interact with it in any of a number of possible ways. Other

neutrons will simply pass it without any interaction.

It is interesting to note that the imaginary projected area or target area of a nucleus, as

shown in Figure 2, may be larger or smaller than the actual projected area as determined

from the physical size of the nucleus. It may be larger because the nucleus has a sphere of

influence surrounding it and any neutron passing within this sphere of influence may be



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attracted to interact with it. It may be smaller because some nuclei may allow neutrons to

pass right through themselves without any interaction taking place. The imaginary

projected area may thus be considered as being related to the probability of a reaction

occurring-the larger the area, the greater the probability of interaction.

It is also interesting to note that for different reactions with the nucleus there are different

degrees of probability of interaction and therefore effectively different imaginary projected

areas. Uranium-238 for example has a larger imaginary target area for elastic scattering

than for radiative capture illustrating that there is a greater probability of elastic scattering

occurring. It is convenient for illustrative purposes to draw a pie diagram with the total

area signifying the probability of all interactions occurring and each slice representing the

probability of individual interactions taking place.

Figure 2. Target areas of nuclei for different reactions

These imaginary projected areas are known as nuclear cross sections and indicate the

probability of any interaction occurring. The cross sections of the nuclei of individual

atoms are measured in square centimeters, square meters or barns where:

1 barn = 1 x 10-24 cm2

1 barn = 1 x 10-28

m2



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If the actual projected area of a nucleus is calculated it is found that for mid-range elements

with an atomic mass number of about 90 this area is equal to 1 barn. Lighter elements have

smaller projected areas and heavier elements larger projected areas.

A cross section of 1 barn indicates immediately that the imaginary target area is roughlyequal to the actual projected area of the nucleus. This allows cross sections to be

visualized. A cross section of several hundred barn indicates that the nucleus has a large

sphere of influence around it while a cross section very much smaller than a barn indicates

that the nucleus allows neutrons to pass through it with practically no chance of an

interaction occurring.

The cross section of Uranium-235 for example is 687 barn whereas its physical cross

section is 1.87 barn. Therefore, with an effective area for neutron interaction so much

bigger than that its actual area, it is "as big as a barn" from a nuclear point of view and

hence the term "barn". The term "barn" was proposed in 1942 by physicists M.G.

Holloway and C.P. Baker as a result of such humorous association of ideas.

There are different types of cross sections, in fact there is one type of cross section for each

type of neutron interaction with the nucleus except for the relatively rare nuclear producing

and nuclear transmutation reactions. These different cross sections may be added to give a

total cross section or probability of reaction as shown in Figure 3. The magnitude of each

slice of the "pie" represents the probability of that type of reaction. The nomenclature for

different cross sections is given below with the different types of interactions:

s = Elastic scattering cross section

i = Inelastic scattering cross section

n, = Radiative capture cross section

a = Absorption cross section

f = Fission cross section

Values for these are tabulated but are often combined into two main types of interactions:

s = Scattering cross section ( s and i )

a = Absorption cross section (

n, and f )

When these are combined they are added together so that the scattering cross-section

includes both elastic and inelastic scattering and the absorption cross-section includes both

radiative capture and fission.



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Figure 3. Cross sections of U-235 for various nuclear reactions

For a particular isotope all the individual microscopic cross sections can be added to give

the total microscopic cross section.

total s i a= + + + . . . . .

Generally however any particular calculation requires the application of a specificmicroscopic cross section only.

2.2. Macroscopic Cross Sections

The macroscopic cross section is the cross section density in a material. It is defined as

the number of nuclei per unit volume multiplied by the microscopic cross section

N .

The units are the inverse of length (cm-1

or m-1

)

N = (4)

This provides a basis for the comparison of different materials. A dense material withnuclei of small cross section would be seen by neutrons to be effectively the same as a rare

material with nuclei of large cross section.

For a single isotope the macroscopic cross section can be determined from the above

equation. This gives the effective cross section density in a pure material and indicates the

probability of a neutron interaction within that material.

If there is a homogeneous mixture of different isotopes the cross section density can be

calculated separately for each and then added to give the total macroscopic cross section.



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a a b b c c. . . . .N N N = + + +

Note that is the number of nuclei or atoms of each element per unit volume in the

material.

N

2.3. Number of Nuclei

The number of nuclei per unit volume in a sample is given by the following equation

where is Avogadro's number,

N

AN M the atomic weight, and the density.

A( / )N N M = (5)

For a material such as water H2O or uranium dioxide UO2 where the elements are bound

together as molecules the number of nuclei of each element needs to be properly accounted

for. For example when calculating the number of nuclei of each element in H2O or UO2

using Avogadro's Number and the molecular weightAN mM (approximately 18 for H2O

and 270 for UO2) the number of molecules is obtained as follows:

A m( / )N N M =

In the case of H2O the number of oxygen atoms is equal to the number of molecules while

the number of hydrogen atoms is double the number of molecules.

In many cases it is required only to know the number of nuclei of a specific isotope in a

mixture of elements. In the case of natural uranium with a mass fraction of U-235 the

number of U-235 nuclei will be:

235 235 A 235 235( / )N N M =

Two assumptions may be made in evaluating the number of nuclei without excessive error:

The molecular or atomic weight may be taken as an integer corresponding with theatomic mass number of each constituent.

Mass ratio (enrichment) and volume ratios (isotopic abundance) may be consideredequal for any single element.

In the case of uranium dioxide UO2 enriched to 3% in U-235 and having a density of

10.5 g/cm3

the number of U-235 nuclei per unit volume is:

235 235 A 235 235

235 A 235 fuel

235 238 235 238 oxygen

( / )

= ( / )

[ (1- ) ] /[ (1- ) ]

= 0.881

N N M

N M f

f M M M M M

=

= + + +



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24

235

21 3

27 3

0.03 (0.6022 10 / 235) 0.881 10.5

0.711 10 nuclei/cm

0.711 10 nuclei/m

N =

=

=

2.4. Reaction Rate

Since the macroscopic cross-section is effectively the material parameter seen by the

neutrons and since neutron flux is effectively the number of neutrons passing through a

given place per unit time it follows then that the reaction rate R between neutrons and

nuclei is given by:

R = (6)

This may also be written as:

R N nv= (7)

This is perfectly logical since the reaction rate R would likely be proportional to the

number of nuclei , the cross-sectionN , the number of neutrons and the velocity of the

neutrons -the greater the number of nuclei and neutrons, the greater the chances of a

reaction. The bigger the effective area (cross section) the more likely a nucleus will

intercept a neutron. The higher the velocity of a neutron the sooner it will meet a nucleus.

n

v

2.5. Summary

The following relationships with units summarize the key factors given above.

2.5.1. Macroscopic cross-section

N = nuclei per unit volume (nuclei/m3)

= microscopic cross-section (m2)

N = (m-1)

2.5.2. Neutron Flux

n = neutrons per unit volume (neutrons/m3)

v = neutron velocity (m/s)

nv = (neutrons/m2s)

2.5.3. Reaction Rate

= neutron flux (neutrons/m2s)

= macroscopic cross-section (m-1)

R = (reactions/m3s)



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3. Neutron Scattering and Capture

3.1. Neutron Attenuation

When a beam of neutrons impinges upon a solid body the neutrons interact with nucleiwithin the body. Those not interacting continue through the body. As the beam progresses

through the body more and more interactions occur and less and less neutrons continue on

through the material. The beam of neutrons diminishes in intensity and is attenuated by the

material as shown in Figure 4.

The decrease in intensity dI over any section of material is proportional to the neutron

beam intensity I, microscopic cross-section of the material , number density of nuclei

and the thickness of the material dx :N

dI I Ndx=

If the macroscopic cross section is used this becomes:

dI I dx=

The solution to this differential equation is:

0

xI I e = (8)

This is the equation for the attenuation of a neutron beam. The attenuation of a -ray beam

is similarly:

0

xI I e = (9)

Here is the attenuation coefficientof the -ray beam.

Figure 4. Neutron attenuation in a material



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3.2. Mean Free Path

It is convenient to express the attenuation of neutrons in terms of the average distance

traveled by a neutron before interacting with a nucleus. This is known as the neutron mean

free path .

If the value for intensity I from the solution of the differential equation is substituted into

the differential equation the following is obtained.

-

0 xdI I e dx=

Neutrons in this beam have traveled a distance x without interacting with any nucleus. For

an infinite slab the total distance traveled by all neutrons is:

00 0

x

xx

x dI I x e dx=

= =

The mean free path is this total interaction divided by the original beam intensity

0 00

/xI x e dx I =

1/ = (10)

The neutron mean free path may also be deduced from the probability of an interaction

and the distance traveled before that interaction as illustrated in Figure 5.

Figure 5. Neutron mean free path



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The reaction rate R is equal to the macroscopic cross section multiplied by the neutron

flux .

R =

R n= v

The reaction rate R can also be written in terms of the number of neutrons n multiplied by

their velocity v and divided by their mean free path .

/R nv =

This in effect states that more reactions will occur when the velocity is higher and the mean

free path lower. If these two equations for reaction rate are combined then the following is

obtained:

/nv nv =

1/ = (11)

Thus the mean free path is the inverse of the macroscopic cross-section .

3.3. Scattering Characteristics

It was seen previously that, with elastic scattering, the neutron rebounded from a nucleus

with kinetic energy conserved and no excitation of the nucleus. Furthermore, with inelastic

scattering, the neutron interacted with the nucleus leaving it in an excited state.

Both of these scattering effects may occur in a single nuclide and it is found that the

probability of these reactions is, to a large degree, dependent upon the energy of the

incoming neutron.

At very low energies, the neutron does not excite the nucleus and is scattered as if

influenced by the physical size of the nucleus which is given in terms of atomic mass

number A by the following:

15 1/3

1.25 10r

= A (m) (12)

This gives the physical cross sectional areas for most nuclei of about 1 barn. The apparent

area of the nucleus for elastic scattering is the neutron scattering cross sections

which

generally ranges from about 4 barn to 12 barn for most elements. This discrepancy

indicates that the neutron itself has a certain physical size and that the nucleons of certain

elements are not necessary closely packed in the nucleus.

This scattering at low neutron energy is calledpotential scattering and is constant over a

range of low neutron energies.

At intermediate energies some neutrons have an energy that raises the nucleus to a discrete

excitation level. Under these conditions absorption and subsequent emission of a neutron



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occurs more easily. Since the nucleus is left in an excited state the emitted neutron is at a

lower energy. This results in inelastic scattering. If there is no match in vibrational

characteristics of the neutron and the nucleus, absorption does not occur easily. However, if

there is a match between the vibrational characteristics, the neutron is readily absorbed

resulting in a lesser chance of scattering. This results in widely varying scatteringprobabilities over a certain range of neutron energies. This is called the resonance region.

At very high energies there is no longer a match in vibrational characteristics and the

probability of scattering falls with increasing energy as those neutrons passing close to the

nucleus are less affected by it. This is known as the smooth region.

These regions and other characteristics are shown in Figure 7.

3.4. Absorption Characteristics

It was seen previously that, with both inelastic scattering and radiative capture, the neutron

interacted with the nucleus leaving it in an excited state. Both of these interactions may

occur in a single nuclide and it is found that the probability of these reactions is to a large

degree dependent upon the energy of the incoming neutron.

For many nuclides there is a threshold neutron energy above which inelastic scattering

occurs and below which radiative capture occurs. This is due to the fact that the neutron

brings with it a certain amount of energy which is transferred to the nucleus when it enters

the nucleus. If the neutron energy is sufficient to raise the energy of the nucleus above the

threshold value then the excited nucleus can emit a neutron along with a -ray. If the

energy of the excited nucleus remains below the threshold value no neutron will appear andonly a -ray will be emitted. The threshold energy corresponds with the binding energy of

the additional neutron while the -ray corresponds with the amount of energy remaining

above the ground state of the nucleus. High velocity (high energy) neutrons are thus likely

to be elastically scattered while low velocity (low energy) neutrons are likely to suffer

radiative capture.

3.5. Radiative Capture Model

From the above it is evident that radiative capture is likely to occur with neutrons below the

threshold energy value, that is, with lower velocity neutrons. As the velocity is decreasedfurther it is found, for many nuclides, that the probability of radiative capture increases.

This probability is in fact inversely proportional to the velocity (square root of energy).

This can be visualized by imagining that the nucleus has a sphere of influence around it as

illustrated in Figure 6.

A neutron passing through this sphere of influence will spend a certain period of time

within that sphere of influence. For a given path, the higher its velocity the shorter the time

spent within the sphere of influence. If the probability of capture is proportional to the time

spent within the sphere of influence, then the probability of capture (absorption cross-

section a ) will be inversely proportional to velocity .v



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a1/v (13)

Figure 6. Interaction probability with respect to neutron velocity.

3.6. Cross Sections

The above may be summarized and illustrated by plotting on a composite diagram as in

Figure 7.

Figure 7. Variation of typical cross sections with neutron energy

The elastic scattering cross-section s is constant in the low energy potential region,



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fluctuates in the resonance region and falls slowly with increasing energy in the smooth

region. The inelastic cross-sectioni

is only apparent above a certain threshold energy.

The radiative capture cross-section

is inversely proportional to velocity in the 1/

region, fluctuates in the resonance region and drops to a low value or disappears at highenergies. The total cross-section

v

t is a summation of all the individual cross-sections

including fission. Note that both the cross-section and neutron energy are plotted on

logarithmic scales.

4. Neutron Moderation

4.1. Neutron Energy Changes

When neutrons interact with nuclei by elastic or inelastic scattering their energy is

degraded. Generally neutrons produced from fission have an energy of about 2 MeV while

neutrons after thermalization have an energy of about 0.025 eV. The number ofinteractions to bring about this degradation depends upon several factors including the

initial energy of the neutrons and the type of scattering (elastic or inelastic). Inelastic

scattering generally requires that the incoming (captured) neutron have sufficient energy to

excite the nucleus to a level that will result in the ejection of a neutron. Hence inelastic

scattering occurs only at high neutron energies and the resulting neutrons will have very

much lower energies. Elastic scattering, on the other hand, occurs at all neutron energies

and may not necessarily degrade the neutron energy very much. Hence, any neutrons

produced from fission that suffer inelastic collisions initially will subsequently be subject

to a series of elastic collisions. Those that suffer elastic collisions initially will also likely

continue to degrade their energy by elastic collisions. Hence most collisions are elastic.

4.2. Logarithmic Mean Energy Decrement

When neutrons interact with nuclei in elastic scattering collisions they lose energy. The

amount of energy lost depends upon the mass of the nucleus and the angle of incidence of

the neutron. More energy is lost when a neutron strikes a light nucleus than when it strikes

a heavy one. Also more energy is lost in a head-on collision than in a glancing collision.

The minimum energy after one collision is:minE

min0E E=

where:

2[( -1) /( 1)]A A = +

Here A is the atomic mass number of the nucleus and it is evident that the higher this

number the closer will be to unity and the smaller the maximum loss in energy.Considering the results of various angles of incidence of the incoming neutron, it is found

that the average energy after one collision is:aveE



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ave 0(1/ 2)(1 )E E= +

The average energy loss after one collision is thus given by:

0 av-E E E = e

0(1 /2)(1 - )E E =

where:

2[( -1) /( 1)]A A = +

Since the loss or change in energy depends upon the incoming neutron energy and since

this is lower in each subsequent collision in an exponential manner, it is convenient to

express the change in energy in logarithmic terms. Furthermore, since the change in energy

is different for each collision, the average of the logarithmic values of the initial energy

and resultant energy0E E is used. The logarithmic mean energy decrement is the

average of the difference of these logarithmic energy values:

0 ave[ln - ln ]E E rage =

0 average[- ln( / )]E E =

The value of the logarithmic mean energy decrement for any isotope of atomic massnumber A is given as follows:

21 [( -1) / 2 ]ln[( -1) /( 1)]A A A A = + + (14)

An approximate value for the logarithmic mean energy decrement is given by the following

empirical equation:

2 /[ (2 / 3)]A = + (15)

This equation has negligible error for all but the very lowest atomic mass numbers hence itis widely adopted in place of the theoretical equation. The number of elastic collisions

required for the neutron energy to drop from an initial energy to a final energy is then

given by:

N

iE fE

i fln( / )N E E = (16)

The value of for a high energy neutron from fission (2 MeV) to become thermalized at

ambient conditions (0.025 eV) is 18 for Hydrogen, 43 for Helium and 115 for Carbon.

Lighter elements are efficient at reducing neutron energy because they are light and absorb

a lot of energy when struck by a nucleus. Collision parameters for a few other common

N



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materials are given in Table 1.

Nucleus Mass

Number

A

Mass Number

Ratio

Mean

LogarithmicEnergy

Decrement

Number of

Collisions toThermalize

N

Hydrogen

H2O

Deuterium

D2O

Helium

Beryllium

BeO

Carbon

Oxygen

Sodium

Iron

Uranium

1

2

4

9

12

16

23

56

238

0

0.111

0.360

0.640

0.716

0.779

0.840

0.931

0.983

1.000

0.920*

0.725

0.509*

0.425

0.209

0.174*

0.158

0.120

0.0825

0.0357

0.00838

18

20

25

36

43

83

105

115

152

221

510

2172

2[( -1) /( 1)]A A = + * An appropriate average value.

Data obtained from Lamarsh and Baratta, Introduction to Nuclear Engineering, Prentice

Hall, 2001

Table 1. Scattering collision parameters of some common materials

4.3. Definitions

It has already been shown that the probability of radiative capture of neutrons by many

nuclides increases as the energy (and hence velocity) of the neutrons is decreased. Theprobability of capture (absorption) is inversely proportional to the neutron velocity over a

range of neutron energies. The fissioning of certain fissile materials such as U-235 is the

result of the absorption of a neutron so it follows that the probability of fission in these

materials will also increase with a reduction in neutron velocity. To enhance the fission

process therefore it is advantageous to reduce the energy of the neutrons to a lower value

by passing them through a suitable material or moderator. This material however should

not absorb neutrons (by radiative capture) too strongly as this would reduce the number of

neutrons available for causing fission.

Materials suitable for slowing down or moderating neutrons without excessive absorption

of them may be assessed by using the following equations and definitions:



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4.3.1. Mean Logarithmic Energy Decrement

i fln( / )N E E =

N = number of collisions

iE= initial energy (2 MeV after fission)

fE = final energy (0.025 eV when thermalized)

4.3.2. Macroscopic Scattering Cross Section s

sN =

s (m-1)

N = nuclei per unit volume (nuclei/m3)

s = microscopic scattering cross section (m

2)

4.3.3. Slowing Down Power

Slowing down power = s (m-1

)

4.3.4. Moderating Ratio

Moderating ratio = s a/

Table 2 gives the above parameters for some materials suitable as moderators.

Moderator Mean

Logarithmic

Energy

Decrement

Macroscopic

Scattering

Cross

Section(a)

s

(cm-1)

Slowing

Down

Power

s

Macroscopic

Absorption

Cross Section

a

(cm-1)

Moderating

Ratio

s a/

He(b)

Be

C(c)

BeOH2O

D2O

D2O

D2O

0.425

0.206

0.158

0.1740.927

0.510

0.510

0.510

2 x 10-6

0.74

0.38

0.691.47

0.35

0.35

0.35

9 x 10-6

0.15

0.06

0.121.36

0.18

0.18

0.18

very small

1.17 x 10-3

0.38 x 10-3

0.68 x 10

-3

22 x 10-3

0.33 x 10-4(d)

0.88 x 10-4(e)

2.53 x 10-4(f)

large

130

160

18060

5500(d)

2047(e)

712(f)

Data obtained from NB Power Nuclear Training Course 22007

(a) Average value for epithermal neutrons (energies between 1 eV and 1000 eV)

(b) At standard temperature and pressure

(c) Reactor-grade graphite

(d) 100% pure D2O



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(e) Reactor-grade D2O (99.75 pure)

(f) 99% pure D2O

Table 2. Slowing down and moderating properties of moderators

5. Fission and Fusion

5.1. Energy Release

It has been shown that both the fusion of light elements and the fission of heavy elements

will produce energy. This is due to the fact that the binding energy per nucleon is less for

light and heavy elements than for mid-range elements. The amount of energy released can

be calculated from the mass defect if the final products are known. For fusion a range of

different reactions is possible as hydrogen fuses into helium. For fission only one reaction

is possible for any particular fuel but a range of fission products is produced. On average

about 200 MeV is produced from a fission reaction. Typical fusion and fission reactionsare shown in Figure 8.

Figure 8. Typical fusion and fission reactions



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5.2. Fission

During the fission process a number of neutrons is released since otherwise the resulting

fission products would have too many neutrons and be too far off the stability range. Even

so they have an excess of neutrons and decay towards a more stable condition. Theseneutrons are free to enter other fissile nuclei and so cause further fissions to maintain a

chain reaction. If the same number of neutrons continues into the next generation the chain

reaction is stable. To achieve this some neutrons must be captured without producing

fission since, for every neutron causing fission, on average two or three are produced.

Figure 9 shows the number of neutrons emitted from the fission of U-235 for different

fission reactions (different fission products).

Figure 9. Prompt neutron emission from U-235 per 100 fissions.

Fission occurs spontaneously in some heavy nuclides but is rare. This contributes to the

gradual decay of the nuclide and creates a few free neutrons within the fuel. This is an

important factor when loading new fuel into a reactor as the resulting low level nuclear

chain reactions could inadvertently grow out of control. Fission induced by neutrons is due

to the fact that the incoming neutron adds sufficient energy to the nucleus to raise its energy

level enough for it to become unstable. Nuclides that fission when unstable are known as

fissile materials. There are four such fissile isotopes:

Uranium-233

Uranium-235

Plutonium-239

Plutonium-241

Thermal neutron interaction parameters for these fissile materials are given in Table 3.



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Nuclide Microscopic

Absorption

Cross

Section

a

Microscopic

Capture

Cross

Section

Microscopic

Fission

Cross

Section

f

Capture

Fission

Ratio

Neutrons

Emitted

per

Absorption

Neutrons

Emitted

per

Fission

U-233

U-235

Pu-239

Pu-241

Natural U

578.8

680.8

1011.3

1377

7.59

47.7

98.6

268.8

368

3.40

531.1

582.2

742.5

1009

4.19

0.090

0.169

0.362

0.365

0.811

2.287

2.068

2.108

2.145

2.24

2.492

2.418

2.871

2.927

3.06

a f= +

f a f/ ( - ) /

f = = f a( / ) =

Data obtained from Lamarsh and Baratta, Introduction to Nuclear Engineering,

Prentice Hall, 2001

Table 3. Thermal neutron (0.025 eV) data for fissile nuclides

A number of other nuclides will fission if the incident neutron has a high kinetic energy.

This kinetic energy together with the binding energy can raise the energy level of the

nucleus sufficiently for it to become unstable and to fission. Such nuclides are known as

fissionable materials. Fissionable isotopes thus require energetic neutrons to cause fission

and as such are nonfissile.

5.3. Fission Characteristics

Uranium-235 and Uranium-238 have scattering and absorption cross-sections similar to

other materials. Refer to Figure 10.

In U-235 absorption usually leads to fission and in the low neutron energy region the

absorption cross-section is very high but decreases with increasing neutron energy since it

is inversely proportional to the neutron velocity. There is then a resonance region where

there are peaks with a high probability of absorption. At high energies there is a low

probability of absorption and hence fission and the cross-section is low. In U-238absorption does not lead to fission except at very high neutron energies. At low neutron

energies there is a low probability of absorption and this is also inversely proportional to

neutron velocity. In the resonance region however there are very high peaks of absorption.

The absorption cross-section then falls again to low values in the high energy region. At

very high energies absorption leads to fission.



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Figure 10. Fission and absorption characteristics of uranium.

5.4. Fission Products

During fission two fission fragments usually of unequal mass are produced. These

generally have atomic mass numbers of between 100 and 140 though a range of

possibilities exists from an atomic mass number of about 70 to about 160 as shown in

Figure 11. The amount of a particular fission product occurring is known as thefission

yield. Fission yields vary for different fissile materials and for fission with higher energy

neutrons. The fission yields of Plutonium-239, for example, show that somewhat more

fission products of intermediate mass number are produced than is the case with Uranium-

235. For high energy neutrons the fission yield curve is much flatter still with even more

fission products of intermediate mass being produced.



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Figure 11. Fission yields for U-235 and Pu-239

5.5. Neutron Energy Spectrum

Neutrons produced at the time of fission are known asprompt neutrons. Some neutrons

appear a short time later and these are known as delayed neutrons. The prompt neutrons

are produced with a range of different energies. Most energy from fission appears as

kinetic energy of the heavy fission products but some is carried away by the neutrons also

as kinetic energy.

Figure 12. Prompt neutron energy distribution per 100 neutrons



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The energy of prompt neutrons varies from about zero to about 8 MeV. If a sample of 100

prompt neutrons is analyzed, as in Figure 12, it is found that some 35 have an energy of

about 1 MeV, the most probable energy, while the average energy is about 2 MeV. The

results are usually plotted as a smooth curve of fraction emitted versus neutron energy.

5.6. Delayed Neutrons

Delayed neutrons are emitted from some fission products a short while after fission has

occurred. Most fission products are unstable and decay towards a more stable state by

emitting particles, usually -particles to convert a neutron into a proton. Some however

are sufficiently unstable to emit neutrons directly or subsequently (after -particle

emission) to reduce the neutron number. An example is the fission product Bromine-87.

This decays to Krypton-87 by the emission of a -particle and then to Krypton-86 by the

emission of a neutron. The half-lives for these reactions are so short that the neutrons

appear almost immediately but the time lag is sufficiently important to have a very markedinfluence on the control of nuclear reactors. The delay is long enough to be detected by

control systems which can respond rapidly enough to changes in delayed neutron

production. No control system can respond rapidly enough to changes in prompt neutron

production.

Delayed neutrons come from some twenty fission products or delayed neutron precursors.

Each precursor produces a neutron following decay or decays of different half-lives. For

convenience these are grouped into six groups of precursors, as shown in Table 4, such that

each group produces neutrons following decay according to a particular half-life. The first

group has a half-life of 55 seconds while the last group has a half-life of only 0.2 second.

Each group has a different yield of neutrons per fission with the fourth group producingnearly 40% while the first and last groups produce only about 3% and 4% respectively.

Overall the total yield of delayed neutrons is only 0.65% of all neutrons produced in

fission. This small amount however is very important in the control of nuclear reactors and

the control system must be able to detect small enough changes in the neutron flux to

maintain control on delayed neutrons.

Group Half life

1/2t (s)

Decay

constant

(s-1)

Relative

Yield

(%)

Yield

(neutrons per

fission)

Fraction

1

2

3

4

5

6

55.72

22.72

6.22

2.30

0.610

0.230

0.0124

0.0305

0.111

0.301

1.14

3.01

3

22

20

39

12

4

0.00052

0.00346

0.00310

0.00624

0.00182

0.00066

0.000215

0.001424

0.001274

0.002568

0.000748

0.000273



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Total 100 0.01580 0.006502

Data obtained from Lamarsh and Baratta, Introduction to Nuclear Engineering,Prentice Hall, 2001

Table 4. Delayed neutron data for thermal fission of U-235

5.7. Fission Process Summary

For fission to occur the incoming neutron must add sufficient energy to the fissile nucleus

to raise its energy above the critical value for fissioning. For the four fissile materials,

thermal neutrons add sufficient binding energy to achieve this. Low energy neutrons

interact more readily with Uranium-235 to cause fission than do high energy neutrons.

Uranium-238 on the other hand will only undergo fission with high energy neutrons. The

shape of the neutron-proton ratio curve results in additional neutrons being produced in

fission. These additional neutrons allow for a chain reaction to be established with

subsequent fissions with each new generation of neutrons. Neutrons produced in fission

have a range of energies with an average of about 2 MeV. These high energy neutrons

must be slowed down or moderatedto reduce their energy so as to be able to interact easily

with further Uranium-235 nuclei to start a new cycle. The energy produced in one fission

process is about 200 MeV made up as tabulated in Table 5. By arranging for multiple

parallel fissions in a continuing controlled chain reaction a steady production of energy can

be achieved.

5.8. Charged Particles

Fission products are produced as a light fragment and a heavy fragment from each fission.

The lighter fragments have kinetic energies of about 100 MeV while the heavier fragments

have energies of about 70 MeV. This division of energies arises from the conservation of

momentum as two initially stationary parts of different mass recoil from one another.

These fission fragments leave behind some twenty electrons and immediately become

positively charged. They lose kinetic energy rapidly in the surrounding material producing

heat and ionization along their path. Their range is very short being in the order of 1.4 x

10-3

cm (0.014 mm) in uranium dioxide fuel (UO2)

Energy Source Recoverable energy

(MeV)



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Fission fragments

lighter fragment (kinetic energy)

heavier fragment (kinetic energy)

Fission product decay-rays-rays

Prompt -raysFission neutrons (kinetic energy)

Capture -rays

100

68

8

6

7

5

6

Total 200

Table 5. Energy produced by the fission of U-235

Alpha particles emitted from heavy nuclides also interact with other atoms causing

ionization. They travel in a short straight path with a range dependent upon their energy

according to the following formulae where is the density and M the molecular weight

air ( )Range function Energy=

1/ 2

medium air air medium medium air( / )( / )Range Range M M =

Beta particles travel in a zigzag path and are not very penetrating since they are very light.

Their range is also a function of their energy

max medium( ) /Range function Energy =

Glossary

Fission: Nuclear reaction involving the splitting of a heavy atom into two

lighter atoms.

Fusion: Nuclear reaction involving the joining of two light atoms into asingle heavier atom

Macroscopic Cross

Section:

Cross section density in material for a specified nuclear reaction

Microscopic Cross

Section:

Effective cross section of nucleus for a specified nuclear reaction

Neutron Flux: Flow of neutrons per unit area per unit time

Nomenclature

A Atomic mass number

b Barn (1 x 10-28

m2

)



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E Energy

aveE Average energy

fE Final energy

iE Initial energy

KEE Kinetic energy

minE Minimum energy

mpE Most probable energy

0E Energy before interaction

f Mass fraction of fuel

I Neutron beam intensity

k Boltzmann constant (13.8 x 10-24 J/K)

m MassM Atomic weight

mM Molecular weight

n Number of neutronsN Number of collisions

N Number of nuclei per unit volume

AN Avogadro's number

r Radius

R Reaction rate

T Absolute temperaturev Neutron velocityV Particle velocity

mp

V Most probable particle velocity

x Material thickness Collision parameter Mass fraction of isotope

Neutron mean free path Attenuation coefficient for -rays Logarithmic energy decrement

Density Microscopic cross section

a Microscopic absorption cross section

s Microscopic scattering cross section

f Microscopic fission cross section

Macroscopic cross section Neutron flux

Bibliography

El-Wakil, M.M. (1993),Nuclear Heat Transport, The American Nuclear Society, Illinois, United States.

[This text gives a clear and concise summary of nuclear and reactor physics before addressing the core

material namely heat generation and heat transfer in fuel elements and coolants].

Foster, A.R., and Wright, R.L. (1983),Basic Nuclear Engineering, Prentice-Hall, Englewood Cliffs, New

Jersey, United States. [This text covers the key aspects of nuclear reactors and associated technologies. It

gives a good fundamental and mathematical basis for the theory and includes equation derivations and worked



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examples].

General Electric (1989), Nuclides and Isotopes, General Electric Company, San Jose, California, United

States. [This reference booklet contains a complete chart of the nuclides with all their properties (structure,

isotopic mass, half-life, decay modes, absorption properties, etc.) as well as explanatory supporting text].

Glasstone, S. (1979), Sourcebook on Atomic Energy, Krieger Publishing Company, Malabar, Florida, UnitedStates (original 1967), Van Nostrand Reinhold, New York, United States (reprint 1979). [This is a classic text

(over 100 000 English copies sold and translated into seven other languages). It covers all aspects of atomic

theory and nuclear physics from the initial development to the first commercial power reactors of several

different types. It is an excellent historical reference].

Glasstone, S. and Sesonske, A. (1994),Nuclear Reactor Engineering, Chapman and Hall, New York, New

York, United States. [This text, now in two volumes, covers all aspects of nuclear physics and nuclear

reactors including safety provisions and fuel cycles].

Knief, R.A. (1992), Nuclear Engineering: Theory and Technology of Commercial Nuclear Power.

Hemisphere Publishing Corporation, Taylor & Francis, Washington DC, United States. [This book gives a

concise summary of nuclear and reactor physics before addressing the core material namely reactor systems,

reactor safety and fuel cycles. It is a good reference for different types of reactors and historical

developments].

Krane, S.K. (1988),Introductory Nuclear Physics, John Wiley and Sons, New York, United States. [This text

provides a comprehensive treatment of various aspects of nuclear physics such as nuclear structure, quantum

mechanics, radioactive decay, nuclear reactions, fission and fusion, subatomic particles, etc.]

Lamarsh, J.R. and Baratta, A.J. (2001),Introduction to Nuclear Engineering, Prentice-Hall, Upper Saddle

River, New Jersey, United States. [This book provides a comprehensive coverage of all aspects of nuclear

physics and nuclear reactors. It has a good descriptive text supported by all necessary mathematical relations

and tabulated reference data.]

Biographical Sketch

Robin Chaplin obtained a B.Sc. and M.Sc. in mechanical engineering from University of Cape Town in 1965and 1968 respectively. Between these two periods of study he spent two years gaining experience in the

operation and maintenance of coal fired power plants in South Africa. He subsequently spent a further year

gaining experience on research and prototype nuclear reactors in South Africa and the United Kingdom and

obtained M.Sc. in nuclear engineering from Imperial College of London University in 1971. On returning and

taking up a position in the head office of Eskom he spent some twelve years initially in project management

and then as head of steam turbine specialists. During this period he was involved with the construction of

Ruacana Hydro Power Station in Namibia and Koeberg Nuclear Power Station in South Africa being

responsible for the underground mechanical equipment and civil structures and for the mechanical balance-of-

plant equipment at the respective plants. Continuing his interests in power plant modeling and simulation he

obtained a Ph.D. in mechanical engineering from Queen=s University in Canada in 1986 and was subsequently

appointed as Chair in Power Plant Engineering at the University of New Brunswick. Here he teaches

thermodynamics and fluid mechanics and specialized courses in nuclear and power plant engineering in the

Department of Chemical Engineering. An important function is involvement in the plant operator and shiftsupervisor training programs at Point Lepreau Nuclear Generating Station. This includes the development of

material and the teaching of courses in both nuclear and non-nuclear aspects of the program.. He has recently

been appointed as Chair of the Department of Chemical Engineering.

To cite this chapter

R.A. Chaplin ,(2007), NUCLEAR INTERACTIONS, in Nuclear Energy Materials and Reactors ,

[Eds.Yassin A. Hassan, Robin A. Chaplin ], inEncyclopedia of Life Support Systems (EOLSS), Developed

under the Auspices of the UNESCO, Eolss Publishers, Oxford ,UK, [http://www.eolss.net]

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